vcaa 2006 mm cas 1

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Victorian Certifi cate of Education2006

MATHEMATICAL METHODS (CAS)Written examination 1

Friday 3 November 2006 Reading time: 9.00 am to 9.15 am (15 minutes) Writing time: 9.15 am to 10.15 am (1 hour)

QUESTION AND ANSWER BOOK

Structure of bookNumber ofquestions

Number of questionsto be answered

Number ofmarks

11 11 40

• Students are permitted to bring into the examination room: pens, pencils, highlighters, erasers, sharpeners, rulers.

• Students are NOT permitted to bring into the examination room: notes of any kind, blank sheets of paper, white out liquid/tape or a calculator of any type.

Materials supplied• Question and answer book of 10 pages, with a detachable sheet of miscellaneous formulas in the

centrefold.• Working space is provided throughout the book.

Instructions• Detach the formula sheet from the centre of this book during reading time.• Write your student number in the space provided above on this page.

• All written responses must be in English.

Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room.

© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2006

2006 MATHMETH & MATHMETH(CAS) EXAM 1 2

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3 2006 MATHMETH & MATHMETH(CAS) EXAM 1

TURN OVER

Question 1Let f (x) = x2 + 1 and g(x) = 2x + 1. Write down the rule of f (g(x)).

1 mark

Question 2For the function f : R → R, f (x) = 3e2x � 1,a. Þ nd the rule for the inverse function f −1

2 marks

b. Þ nd the domain of the inverse function f −1.

1 mark

InstructionsAnswer all questions in the spaces provided.A decimal approximation will not be accepted if an exact answer is required to a question.In questions where more than one mark is available, appropriate working must be shown.Unless otherwise indicated, the diagrams in this book are not drawn to scale.


Question 3a. Let f (x) = ecos (x). Find f ′(x)

1 mark

b. Let y = x tan (x). Evaluate dydx

when x = π6

.

3 marks

Question 4

For the function f :[ −π , π ] → R, f (x) = 5 23

cos x +

π

a. write down the amplitude and period of the function

2 marks

b. sketch the graph of the function f on the set of axes below. Label axes intercepts with their coordinates. Label endpoints of the graph with their coordinates.

3 marks

y

xO π– π

6

5

4

3

2

1

–1

–2

–3

–4

–5

–6


TURN OVER

Question 5Let X be a normally distributed random variable with a mean of 72 and a standard deviation of 8. Let Z be the standard normal random variable. Use the result that Pr(Z < 1) = 0.84, correct to two decimal places, to Þ nda. the probability that X is greater than 80

1 mark

b. the probability that 64 < X < 72

1 mark

c. the probability that X < 64 given that X < 72.

2 marks


Question 6The probability density function of a continuous random variable X is given by

f x

x x( ) =

≤ ≤

12

1 5

0 otherwise

a. Find Pr (X < 3).

2 marks

b. If Pr (X ≥ a) = 58 , Þ nd the value of a.

2 marks


TURN OVER

Question 7The graph of f : [�5, 1] → R where f (x) = x3 + 6x2 + 9x is as shown.

a. On the same set of axes sketch the graph of y = f x( ) .

2 marks

b. State the range of the function with rule y = f x( ) and domain [�5, 1].

1 mark

Question 8A normal to the graph of y x= has equation y = � 4x + a, where a is a real constant. Find the value of a.

4 marks

–3 Ox

y


Question 9A rectangle XYZW has two vertices, X and W, on the x-axis and the other two vertices, Y and Z, on the graph of y = 9 � 3x2, as shown in the diagram below. The coordinates of Z are (a, b) where a and b are positive real numbers.

a. Find the area, A, of rectangle XYZW in terms of a.

1 mark

b. Find the maximum value of A and the value of a for which this occurs.

3 marks

y

x

Y(–a, b) Z(a, b)

X WO


TURN OVER

Question 10Jo has either tea or coffee at morning break. If she has tea one morning, the probability she has tea the next morning is 0.4. If she has coffee one morning, the probability she has coffee the next morning is 0.3. Suppose she has coffee on a Monday morning. What is the probability that she has tea on the following Wednesday morning?

3 marks

CONTINUED OVER PAGE


Question 11Part of the graph of the function f : R → R, f (x) = �x2 + ax + 12 is shown below. If the shaded area is 45 square units, Þ nd the values of a, m and n where m and n are the x-axis intercepts of the graph of y = f (x).

5 marks

END OF QUESTION AND ANSWER BOOK

y

xn mO 3

12

MATHEMATICAL METHODS AND MATHEMATICAL METHODS (CAS)

Written examinations 1 and 2

FORMULA SHEET

Directions to students

Detach this formula sheet during reading time.

This formula sheet is provided for your reference.

© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2006

cheninb

MATH METH & MATH METH (CAS) 2

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3 MATH METH & MATH METH (CAS)

END OF FORMULA SHEET

Mathematical Methods and Mathematical Methods CAS Formulas

Mensuration

area of a trapezium: 12

a b h+( ) volume of a pyramid: 13

Ah

curved surface area of a cylinder: 2π rh volume of a sphere: 43

3π r

volume of a cylinder: π r2h area of a triangle: 12

bc Asin

volume of a cone: 13

2π r h

Calculusddx

x nxn n( ) = −1

x dx

nx c nn n=

++ ≠ −+∫

11

11 ,

ddx

e aeax ax( ) = e dx a e cax ax= +∫1

ddx

xxelog ( )( ) = 1

1x

dx x ce= +∫ log

ddx

ax a axsin( ) cos( )( ) = sin( ) cos( )ax dx a ax c= − +∫1

ddx

ax a axcos( )( ) −= sin( ) cos( ) sin( )ax dx a ax c= +∫

1

ddx

ax aax

a axtan( )( )

( ) ==cos

sec ( )22

product rule: ddx

uv u dvdx

v dudx

( ) = + quotient rule: ddx

uv

v dudx

u dvdx

v

=−2

chain rule: dydx

dydu

dudx

= approximation: f x h f x h f x+( ) ≈ ( ) + ′ ( )

ProbabilityPr(A) = 1 – Pr(A′) Pr(A ∪ B) = Pr(A) + Pr(B) – Pr(A ∩ B)

Pr(A|B) =Pr

PrA B

B∩( )

( )mean: µ = E(X) variance: var(X) = σ 2 = E((X – µ)2) = E(X2) – µ2

probability distribution mean variance

discrete Pr(X = x) = p(x) µ = ∑ x p(x) σ 2 = ∑ (x – µ)2 p(x)

continuous Pr(a < X < b) = f x dxa

b( )∫ µ =

−∞

∞∫ x f x dx( ) σ µ2 2= −

−∞

∞∫ ( ) ( )x f x dx

vcaa 2006 mm cas 1

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